Abstract
In order to research the influence of the control delay time on the vibration isolation system, the dynamic characteristic analysis of a twodegreeoffreedom vibration system, based on highstaticlowdynamicstiffness vibration isolator (HSLDSVI) with timedelayed feedback control is presented. The twodegreeoffreedom nonlinear vibration isolation dynamical system model comprising the delay time parameter is also derived. Then, the effects of feedback gain coefficient and delay parameter on the dynamic characteristics are analyzed with multiple scales method. The results demonstrate that the amplitude can be suppressed by the timedelay variation for a fixed feedback gain. The system can exhibit periodic motion or chaotic vibration when the feedback gain coefficient and timedelay parameter are adjusted, which is beneficial to the application of highstaticlowdynamicstiffness vibration isolator.
Highlights
 In order to research the influence of the control delay time, the mathematical model of the twodegreeoffreedom nonlinear vibration system based on the HSLDSVI is established
 The influence of timedelayed feedback on the system vibration amplitude is investigated by multiple scales method, and the system has different responses with the increase of the time delay
 The analytical solution is in good agreement with the numerical results, the system exhibit periodic or chaotic vibration with the parameters, which is beneficial to the application of the HSLDSVI
1. Introduction
How to suppress lowfrequency vibration effectively has always been a concerned issue in the field of vibration control. In general, to increase the vibration isolation range, the system stiffness should be reduced at the same time, but this will reduce its loading capacity. To overcome this contraction, many scholars have proposed an isolator with highstaticlowdynamicstiffness. The highstaticlowdynamicstiffness vibration isolator (HSLDSVI) is comprised of a positive and a negative stiffness structure. The former part is used to carry the main load, and the latter part is used to offset the stiffness of the positive mechanism, so that the system stiffness could equal to zero at the equilibrium position. Kovacic and Carrella et al. [1, 2] designed an HSLDSVI by connecting two oblique springs as negative stiffness, and its amplitudefrequency and transmission characteristics were further studied in detail. Xu and Zhou et al. [3, 4] designed an HSLDSVI by using the electromagnetic negative stiffness spring and conducted the theoretical and experimental analysis. Meng et al. [5] put forward to design HSLDSVI with equal thickness and variable thickness butterfly spring, and the parameters influence on the transmission rate was studied as well. Huang [6] used the Euler buckling beam with negative stiffness to analyze the static and dynamic characteristics, the experimental results showed that the HSLDSVI had a wider vibration isolation band than the linear isolator.
Meanwhile, the timedelay parameter has a direct influence on mechanical vibration feedback control. Hu et al. [7] studied the primary resonance and 1/3 harmonic resonance of the Duffing system with delayed state feedback method and applied at multiple scales methods to study the required gain and delay parameter values for effective vibration control. Olgac [8] proposed a timedelay resonator in a coupled linear forced vibration system to eliminate the vibration and obtained the theoretical basis and experimental results in timedelay vibration reduction. Maccari [9] studied the delayed feedback control method for suppressing largevalued response and quasiperiodic motion in Van der Pol oscillators and implemented this method to the control of the main resonance of a cantilever beam. Besides, Hosek [10] used a centrifugal pendulum timedelay resonator to control the torsional system, the vibration was removed by adjusting the feedback gain and timedelay online. Chatterjee [11] studied the acceleration delay feedback control method in a nonlinear mechanical vibration system. It was found that the delay acceleration feedback also affected the system's damping characteristics. Cheng [12] studied the HSLDSVI dynamic analysis with the timedelayed feedback, results showed the appropriate feedback parameters could improve the vibration isolation performance.
At present, the influence of the controlling delay time on twodegreeoffreedom system with a highstaticlowdynamicstiffness isolator is expected to be studied in detail. This paper establishes a mathematical model of multidegreeoffreedom highstaticlowdynamic stiffness vibration system with timedelayed feedback control, and the dynamic characteristics are also researched under different delays and feedback gain parameters.
2. Static analysis of HSLDSVI
The HSLDSVI schematic is shown in Fig. 1. Two oblique springs as negative stiffness and a vertical spring as positive stiffness are connected at point $O$. The positive and negative stiffness are ${k}_{v}$${k}_{l}$ respectively, and the parameters $a$ and ${h}_{0}$ are defined as the system geometry, the coordinate x defines the displacement from the position $O$.
Fig. 1the HSLDSVI schematic
The expression between force $f$ and displacement $x$ is given by:
Introducing the nondimensional parameters:
Eq. (1) can be rewritten as:
Differentiating Eq. (2), the nondimensional stiffness $\widehat{K}$ can be given:
Let $\widehat{x}=h$ and $\widehat{K}=0$, the conditions with the system zero stiffness can be obtained:
The nondimensional force and stiffness for three conditions when the parameter $\alpha =\text{1}$ are shown in Fig. 2. It can be seen that the system stiffness at the static equilibrium position is zero when the parameter $h={h}_{QZS}$, but the stiffness is always positive in the other range. If the parameter $h<{h}_{QZS}$, the system will exhibit positivestiffness continuously. Otherwise, if the parameter $h>{h}_{QZS}$, the system stiffness will be negative at the static equilibrium position.
Fig. 2Nondimensional forcedisplacement and stiffnessdisplacement curves of the HSLDSVI (α=1)
Fig. 3Comparison between the approximate and exact expressions of nondimensional force and stiffness (α=1, h=hQZS)
When the displacement is small, Eq. (2) can be expanded by using a Taylor series at the static equilibrium position $\widehat{x}=h$ and let $\widehat{y}=\widehat{x}h$, we can get:
where, $\delta =1+2\alpha \frac{2\alpha}{\sqrt{1{h}^{2}}}$, $\beta =\frac{\alpha}{{\left(1{h}^{2}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{1ex}{$2$}\right.}}$. The nondimensional stiffness can be obtained by differentiating Eq. (6):
The comparison between approximate and exact expressions of nondimensional force and stiffness is illustrated in Fig. 3. The error between the two curves increases with the increase of displacement. When the amplitude of displacement at the static equilibrium position is small, the error can be taken into account. So the approximate expression can be used to imitate the exact expression.
3. Analysis by multiple scales method
The twodegreeoffreedom system with timedelayed feedback is illustrated in Fig. 4. It includes an HSLDSVI between the upper mass ${M}_{1}$ and middle mass ${M}_{2}$, the linear stiffness spring ${K}_{2}$ between the middle mass and foundation. where, ${Z}_{1}$ and ${Z}_{2}$ are the vibration displacement, ${C}_{1}$ and ${C}_{2}$ are the system damping, the excitation force amplitude is $F$, and its excitation frequency is $\mathrm{\Omega}$. In addition, the actuator is installed between the upper and middle mass, so the control force can be expressed as $P={K}_{1}{Z}_{1\tau}$.
Fig. 4Twodegreeoffreedom nonlinear vibration system with timedelayed feedback control
Thus, the system equations can be obtained according to Newton’s law of motion:
Substitute Eq. (5) into Eq. (7), and introduce the nondimensional parameters:
The system nondimensional equation can be get:
To analyze the main resonance by the multiple scales method, it is assumed that:
Where, $0<\epsilon \le 1$. The forms of twoscale approximate expansion in Eq. (9) are assumed as:
Where, ${T}_{n}={\epsilon}^{n}t$, $(n=\mathrm{0,1})$. Using the following differential equations:
where, ${D}_{k}=\partial /\partial {T}_{k}$, ($k=$ 0, 1). Substituting Eqs. (9)(14) into Eq. (8), and equating the coefficients of same power $\epsilon $. If the power $\epsilon $ is zero, we can get:
If the power $\epsilon $ is one, we also can get:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\xi}_{1}{D}_{0}\left({z}_{10}({t}_{0},{t}_{1}){z}_{20}({t}_{0},{t}_{1})\right)k{z}_{10\tau}({t}_{0},{t}_{1})\gamma {z}_{10}^{3}({t}_{0},{t}_{1})$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+3\gamma {z}_{20}({t}_{0},{t}_{1}){z}_{10}^{2}({t}_{0},{t}_{1})3\gamma {z}_{20}^{2}\left({t}_{0},{t}_{1}\right){z}_{10}\left({t}_{0},{t}_{1}\right)+\gamma {z}_{20}^{3}\left({t}_{0},{t}_{1}\right)+{f}_{0}\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right),$
The general solution of Eq. (15) and Eq. (16) can be set as:
where, $A\left({t}_{1}\right)$ and $B\left({t}_{1}\right)$ are undetermined functions, $cc$denotes the complex conjugate term. $C\left({t}_{1}\right)=\frac{{v}_{1}^{2}}{{v}_{1}^{2}{v}_{2}^{2}}A\left({t}_{1}\right)$. The external excitation and time delay can be expressed as the following complex form:
Assuming $\tau $ and $\epsilon $ are very small, ${B}_{\tau}\left({t}_{1}\right)$ and ${C}_{\tau}\left({t}_{1}\right)$ can be expanded by using Taylor series:
Substitute Eqs. (19)(24) into Eq. (17) and Eq. (18), we can obtain:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+3\gamma A\left({t}_{1}\right){B}^{2}\left({t}_{1}\right){e}^{i(2{v}_{1}+{v}_{2}){t}_{0}}3\gamma {A}^{2}\left({t}_{1}\right)B\left({t}_{1}\right){e}^{i({v}_{1}+2{v}_{2}){t}_{0}}+3\gamma A\left({t}_{1}\right){C}^{2}\left({t}_{1}\right){e}^{3i{v}_{2}{t}_{0}}$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}3\gamma {A}^{2}\left({t}_{1}\right)C\left({t}_{1}\right){e}^{3i{v}_{2}{t}_{0}}3\gamma {B}^{2}\left({t}_{1}\right)C\left({t}_{1}\right){e}^{i(2{v}_{1}+{v}_{2}){t}_{0}}+{D}_{0}{\xi}_{1}\left(A\right({t}_{1})C({t}_{1}\left)\right){e}^{i{v}_{2}{t}_{0}}$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+\gamma \left({A}^{3}\right({t}_{1}){C}^{3}({t}_{1}\left)\right){e}^{3i{v}_{2}{t}_{0}}3\gamma B\left({t}_{1}\right){C}^{2}\left({t}_{1}\right){e}^{i({v}_{1}+2{v}_{2}){t}_{0}}kB\left({t}_{1}\right){e}^{i{v}_{1}({t}_{0}\tau )}$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\gamma {B}^{3}\left({t}_{1}\right){e}^{3i{v}_{1}{t}_{0}}2{D}_{0}{D}_{1}C\left({t}_{1}\right){e}^{i{v}_{2}{t}_{0}}{D}_{0}({\xi}_{1}+2{D}_{1})B\left({t}_{1}\right){e}^{i{v}_{1}{t}_{0}}kC\left({t}_{1}\right){e}^{i{v}_{2}({t}_{0}\tau )}$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+\frac{1}{2}{f}_{0}{e}^{i{\sigma}_{0}{t}_{1}+i{v}_{1}{t}_{0}}+cc,$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mu A\left({t}_{1}\right){e}^{i{v}_{2}{t}_{0}}+\mu B\left({t}_{1}\right){e}^{i{v}_{1}{t}_{0}}+\mu c\left({t}_{1}\right){e}^{i{v}_{2}{t}_{0}}+cc.$
Considering the dynamic behavior when the primary resonance and the 1:1 internal resonance occur simultaneously. The deviation values are represented by the introduction of harmonic parameters ${\sigma}_{0}$ and ${\sigma}_{1}$. The ${\sigma}_{0}$ and ${\sigma}_{1}$ denotes the external and internal resonance deviation value respectively:
The solvability conditions for Eq. (23) and (24) can be obtained:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}3\gamma {B}^{2}\left({t}_{1}\right)\stackrel{}{A}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}+6\gamma A\left({t}_{1}\right)C\left({t}_{1}\right)\stackrel{}{A}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}6\gamma B\left({t}_{1}\right)C\left({t}_{1}\right)\stackrel{}{A}\left({t}_{1}\right)$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}3\gamma {C}^{2}\left({t}_{1}\right)\stackrel{}{A}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}+3\gamma {A}^{2}\left({t}_{1}\right)\stackrel{}{B}\left({t}_{1}\right){e}^{2i{\sigma}_{1}{t}_{1}}6\gamma A\left({t}_{1}\right)B\left({t}_{1}\right)\stackrel{}{B}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+3\gamma {B}^{2}\left({t}_{1}\right)\stackrel{}{B}\left({t}_{1}\right)6\gamma A\left({t}_{1}\right)C\left({t}_{1}\right)\stackrel{}{B}\left({t}_{1}\right){e}^{2i{\sigma}_{1}{t}_{1}}+6\gamma B\left({t}_{1}\right)C\left({t}_{1}\right)\stackrel{}{B}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+3\gamma {C}^{2}\left({t}_{1}\right)\stackrel{}{B}\left({t}_{1}\right){e}^{2i{\sigma}_{1}{t}_{1}}+3\gamma {A}^{2}\left({t}_{1}\right)\stackrel{}{C}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}6\gamma A\left({t}_{1}\right)B\left({t}_{1}\right)\stackrel{}{C}\left({t}_{1}\right)$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+3\gamma {B}^{2}\left({t}_{1}\right)\stackrel{}{C}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}6\gamma A\left({t}_{1}\right)C\left({t}_{1}\right)\stackrel{}{C}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}+6\gamma B\left({t}_{1}\right)C\left({t}_{1}\right)\stackrel{}{C}\left({t}_{1}\right)$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+3\gamma {C}^{2}\left({t}_{1}\right)\stackrel{}{C}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}i{v}_{2}{\xi}_{1}A\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}+i{v}_{2}{\xi}_{1}B\left({t}_{1}\right)$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+i{v}_{2}{\xi}_{1}C\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}+2i{v}_{1}B\mathrm{\text{'}}\left({t}_{1}\right)+2i{v}_{2}C\mathrm{\text{'}}\left({t}_{1}\right){e}^{i{\sigma}_{1}{t}_{1}}\frac{1}{2}{f}_{0}{e}^{i{\sigma}_{0}{t}_{1}}=0,$
The general solution of Eq. (29) and Eq. (30) can be written as the followings:
Substituting Eq. (31)(32) into Eq. (29) and Eq. (30), and separating the real part from the imaginary part. We can obtain:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+4{f}_{0}({v}_{1}^{2}{v}_{2}^{2}{)}^{3}\mathrm{s}\mathrm{i}\mathrm{n}({\varphi}_{1})+b({v}_{1}^{2}{v}_{2}^{2}\left)\right({v}_{2}^{4}\left(4k\mathrm{s}\mathrm{i}\mathrm{n}\right({v}_{1}\tau )3\gamma {a}^{2}\mathrm{s}\mathrm{i}\mathrm{n}(2{\varphi}_{2})4{v}_{2}{\xi}_{1})$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+4{v}_{1}^{4}\left(k\mathrm{s}\mathrm{i}\mathrm{n}\right({v}_{1}\tau ){v}_{2}{\xi}_{1})+8{v}_{1}^{2}{v}_{2}^{2}(k\mathrm{s}\mathrm{i}\mathrm{n}({v}_{1}\tau )+{v}_{2}{\xi}_{1}))$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+4a({v}_{1}^{2}+{v}_{2}^{2})({\xi}_{1}{v}_{2}^{5}\mathrm{c}\mathrm{o}\mathrm{s}({\varphi}_{2})k{v}_{1}^{4}\mathrm{s}\mathrm{i}\mathrm{n}({v}_{2}\tau {\varphi}_{2})$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+{v}_{2}{\xi}_{2}\mathrm{c}\mathrm{o}\mathrm{s}\left({\varphi}_{2}\right)+{v}_{1}^{2}{v}_{2}^{2}\left(\mu \mathrm{s}\mathrm{i}\mathrm{n}\right({\varphi}_{2})+k\mathrm{s}\mathrm{i}\mathrm{n}({v}_{2}\tau {\varphi}_{2})+{v}_{2}({\xi}_{1}+{\xi}_{2}\left)\mathrm{c}\mathrm{o}\mathrm{s}\right({\varphi}_{2}\left)\right)\left)\right],$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+3\gamma {a}^{2}{v}_{2}^{6}\mathrm{c}\mathrm{o}\mathrm{s}\left({\varphi}_{2}\right)+b({v}_{1}^{2}{v}_{2}^{2})\left(4{v}_{1}^{4}\right(\mu +k\mathrm{c}\mathrm{o}\mathrm{s}\left({v}_{1}\tau \right))4{v}_{1}^{2}{v}_{2}^{2}(\mu +2k\mathrm{c}\mathrm{o}\mathrm{s}\left({v}_{1}\tau \right))$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+\left(3\gamma {a}^{2}\right(2+\mathrm{c}\mathrm{o}\mathrm{s}\left(2{\varphi}_{2}\right))+4k\mathrm{c}\mathrm{o}\mathrm{s}({v}_{1}\tau \left)\right){v}_{2}^{2})+3\gamma {b}^{3}({v}_{1}^{2}{v}_{2}^{2}{)}^{3}$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+k{v}_{1}^{4}\mathrm{c}\mathrm{o}\mathrm{s}({v}_{2}\tau +{\varphi}_{2})+4a({v}_{1}^{2}{v}_{2}^{2})\left({\xi}_{1}{v}_{2}^{5}\mathrm{s}\mathrm{i}\mathrm{n}\right({\varphi}_{2})+{v}_{2}{\xi}_{2}\mathrm{s}\mathrm{i}\mathrm{n}({\varphi}_{2})$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{v}_{1}^{2}{v}_{2}^{2}(\mu \mathrm{c}\mathrm{o}\mathrm{s}({\varphi}_{2})+k\mathrm{c}\mathrm{o}\mathrm{s}({v}_{2}\tau +{\varphi}_{2})+{v}_{2}({\xi}_{1}+{\xi}_{2}\left)\mathrm{s}\mathrm{i}\mathrm{n}\right({\varphi}_{2}\left)\right)\left)\right],$
where, ${\varphi}_{1}={\sigma}_{0}{t}_{1}{\theta}_{2}$, ${\varphi}_{2}={\sigma}_{1}{t}_{1}+{\theta}_{1}{\theta}_{2}$.
4. Equilibrium solution and stability analysis
The system steadystate motion corresponds to the equilibrium solution of the equation $a\mathrm{\text{'}}\left({t}_{1}\right)=b\mathrm{\text{'}}\left({t}_{1}\right)={\varphi}_{1}^{\mathrm{\text{'}}}\left({t}_{1}\right)={\varphi}_{2}^{\mathrm{\text{'}}}\left({t}_{1}\right)=0$. To further analyze the stability of the equilibrium solution, a perturbation analysis is performed on the Eqs. (33)(36). The characteristic equation for the equilibrium solution can be obtained:
where, ${\lambda}_{i}$ represents the eigenvalues of the Jacobian matrix, ${\rho}_{1}$, ${\rho}_{2}$, ${\rho}_{3}$ and ${\rho}_{4}$ are the coefficient of the characteristic equation. Using RothHurwitz criterion to judge the stability of the equation, and the necessary and sufficient conditions of the system stability are obtained:
A sufficient and necessary condition for the stability of the equilibrium solution in Eq. (8) is the case that the real parts of all the characteristic roots in Eq. (35) are negative. If one of them is positive, then the equilibrium solution of Eq. (8) is not stable. If the real part of the characteristic root changes from the negative to the positive number, the system may have to saddle node bifurcation, which will lead to the emergence of jumping phenomenon. In addition, if the real part of a pair of complex eigenvalues changes from the negative to the positive number, Hopf bifurcation may occur in the system, which will result in complex periodic motions.
5. Simulation analysis
When the parameters ${v}_{1}=\text{1}$, ${v}_{2}=\text{1.2}$, $\mu =\text{0.5}$, ${\xi}_{1}=\text{0.2}$, ${\xi}_{2}=\text{0.2}$, $\gamma =\text{0.5}$, ${f}_{0}=\text{1}$, the amplitude with timedelayed curves are shown in Fig. 5. When the feedback gain is constant, the system vibration amplitude fluctuates with the time delay. It can be observed that the vibration amplitude can be reduced by adjusting the delay time. When the feedback gain coefficient is small, the system steadystate solution is a stable solution. Even if the system takes a large delay, the system remains stable at the equilibrium solution. When the feedback gain increases to 0.2, the system will have unstable solutions in some areas as the delay changes. This means that complex motions such as quasiperiodic motion and chaotic motion will appear in some regions. At the same time, it can be seen from Fig. 6(a) that the system vibration amplitude decreases as the feedback gain increases when $\tau \in $ [3, 8]. As the feedback gain is further increased, the system will have multiple stable solutions when some delay values are taken (for example $\tau =\text{9}$). This implies that the system will exhibit complex movements.
Fig. 5Amplitude and timedelayed curves with different feedback gains
a)
b)
Fig. 6Time response curves with several values of feedback gain and delay
a)$k=\text{0.1}$, $\tau =\text{3}$
b)$k=\text{0.2}$, $\tau =\text{2.7}$
c)$k=\text{0.3}$, $\tau =\text{9}$
d)$k=\text{0.4}$, $\tau =\text{9}$
To illustrate the reliability of the above results, numerical verification of the results in Fig. 5 is conducted. The system time response curves are shown in Fig. 6, and the phase diagrams are shown in Fig. 7. The initial condition is set as ${z}_{1}\left(0\right)={z}_{2}\left(0\right)=$ 0.0001, ${\dot{z}}_{1}\left(0\right)={\dot{z}}_{2}\left(0\right)=$ 0.0001. The response characteristic curves are shown in Fig. 6(a) and Fig.7(a) when the feedback gain is 0.1 and the delay is 3, and the system exhibits a stable periodic motion. It can be seen from Fig. 6(b) and Fig. 7(b) that the system is chaotic at this time when the feedback gain is 0.2 and the delay is 2.7. As is shown in Figs. 6(c)(d) and Figs. 7(c)(d) when the feedback gain is 0.3, 0.4 and the delay is 9, the system shows multiple periodic motions. The numerical results are compared with Fig. 5 and it is found that the numerical simulation results are consistent with the analytical results, which indicates that the conclusions obtained in this paper are reliable.
Fig. 7Phase diagrams with several values of feedback gain and delay
a)$k=\text{0.1}$, $\tau =\text{3}$
b)$k=\text{0.2}$, $\tau =\text{2.7}$
c)$k=\text{0.3}$, $\tau =\text{9}$
d)$k=\text{0.4}$, $\tau =\text{9}$
6. Conclusions
In this paper, the mathematical model of the twodegreeoffreedom nonlinear vibration system based on a highstaticlowdynamicstiffness isolator with timedelayed feedback is established. When the equations contain a delay term, the system exhibits more complex dynamic characteristics due to the influence of nonlinear factors. The influence of timedelayed feedback on the system vibration amplitude is investigated by using multiple scales method, and the amplitude and timedelayed curves are also obtained for the twodegreesof freedom vibration system. It is observed that with the increase in time delay, the system has different responses. Through numerical analysis, it is found that the analytical solution is in good agreement with the numerical results. When the feedback gain coefficient is small, the system will be in a stable periodic motion. For a fixed feedback gain coefficient, the system vibration can be controlled by adjusting the time delay. With the increase of the feedback gain coefficient, the system will exhibit complex multipleperiodic and chaotic motion in certain timedelay regions.
References

Kovacic I., Brennan M. J., Lineton B. Effect of a static force on the dynamic behavior of a harmonically excited quasizero stiffness system. Journal of Sound and Vibration, Vol. 325, Issues 45, 2009, p. 870883.

Carrella A., Brennan M. J., Kovacic I., et al. On the force transmissibility of a vibration isolator with quasizerostiffness. Journal of Sound and Vibration, Vol. 322, Issue 4, 2009, p. 707717.

Xu D., Yu Q., Zhou J., et al. Theoretical and experimental analyses of a nonlinear magnetic vibration isolator with quasizerostiffness characteristic. Journal of Sound and Vibration, Vol. 332, Issue 14, 2013, p. 33773389.

Zhou N., Liu K. A tunable highstaticlowdynamic stiffness vibration isolator. Journal of Sound and Vibration, Vol. 329, Issue 9, 2010, p. 12541273.

Meng L. S., Sun J. G., Wu W. J. Theoretical design and characteristics analysis of a quasizero stiffness isolator using a disk spring as negative stiffness element. Shock and Vibration, Vol. 4, Issue 2015, 2015, p. 119.

Huang X. H., Liu X. T., Sun J. Y., et al. Vibration isolation characteristics of a nonlinear isolator using Euler buckled beam as negative stiffness corrector: A theoretical and experimental study. Journal of Sound and Vibration, Vol. 333, Issue 4, 2014, p. 11321148.

Hu H., Dowell E. H., Virgin L. N. Resonances of a harmonically forced Duffing oscillator with time delay state feedback. Nonlinear Dynamics, Vol. 15, Issue 4, 2004, p. 311327.

Olgac N., Jalili N. Model analysis of flexible beams with delayed resonator vibration absorber: theory and experiments. Journal of Sound and Vibration, Vol. 218, Issue 2, 1998, p. 307331.

Maccari A. Vibration control for the primary resonance of a cantilever beam by a time delay state feedback. Journal of Sound and Vibration, Vol. 259, Issue 2, 2003, p. 241251.

Hosek M., Elmali H., Olgac N. A tunable torsional vibration absorber: the centrifugal delayed resonator. Journal of Sound and Vibration, Vol. 205, Issue 2, 1997, p. 151165.

Chatterjee S. Vibration control by recursive timedelayed acceleration feedback. Journal of Sound and Vibration, Vol. 317, Issue 1, 2008, p. 6790.

Cheng C., Li S. M., Wang Y., Jiang X. X. Performance analysis of highstaticlowdynamic stiffness vibration isolator with timedelayed displacement feedback. Journal of Central South University, Vol. 24, Issue 10, 2017, p. 22942305.
About this article
This research is also funded by the Fundamental Research Funds for the Central Universities (Grant No. 201965006), the project ZR2019QEE031 supported by Shandong Provincial Natural Science Foundation, the State Key Laboratory of Ocean Engineering(Shanghai Jiao Tong University) (Grant No. 1714) and the National Natural Science Foundation of China (Grant No. 51909267, Grant No. 51579242 and Grant No. 51509253).
Haipeng Zhang carried out the static analysis of HSLDSVI. Lihua Yang carried out the modeling and analysis of the vibration isolation system with time delayed feedback control. Pan Su conducted research on the model analysis by multiple scales method. Shuyong Liu mainly explored the equilibrium solution and stability analysis. Bo Zhao analyzed numerical simulation results and prepared the research report.